A Few More Geometries after Ramanujan

We do not know if Ramanujan had any plane curves in mind when working with the integral period functions , and [1], though certainly these imply many simple geometries of intrinsic relevance [2]. This Demonstration gives three additional models characterized by real period functions

.

A new and improved integral-differential algorithm is used to prove the integral periods. Given the input of a well-chosen Hamiltonian , the certified algorithm quickly derives that the associated period function satisfies a special case of the hypergeometric differential equation

.

In prospectus, the functions HEllipticPFDE[Hqp] and DihedralPFDE[Hλϕ] defined in this Demonstration return valid results on infinitely many more calls. These algorithms help to extend the regime of easy integral-differential calculations far beyond the three simple examples depicted here (see Details).

THINGS TO TRY

SNAPSHOTS

DETAILS

Our initial integral-differential algorithm from the Demonstration "Deriving Hypergeometric Picard–Fuchs Equations" produces results, but falls short of satisfying design goals for quality and ease of use. A subsequent Demonstration, "Approximating Pi with Trigonometric-Polynomial Integrals", gives another integral-differential algorithm that automates degree bounding and thus operates on a minimal input. The current Demonstration uses two similar functions, HEllipticPFDE[Hqp] and DihedralPFDE[Hλϕ]. Again our approach avoids general Griffiths–Dwork reduction, but otherwise follows the basic insight of [3, 4]. In this latest version, we automate degree bounding, improve variable handling and reap many new results, all with faster times.

A Hyperelliptic Case

For the more trivial function HEllipticPFDE[Hqp], inputs must have a hyperelliptic form,

with canonical coordinates such that

,

as follows from Hamilton's equations of motion. When the potential reaches a minimum,

,

the phase curves become loops. Each loop is measured by a period function

.

In fact, the invariant differential, , can be integrated on any loop around the Riemannian surface . Then we will obtain a set of period functions that satisfy an easily calculable ordinary differential equation, also called the Picard–Fuchs equation. Let vector collect the one-forms,

,

in order of increasing . Next assume that Hermite reduction follows a particular pattern,

,

with degree-bounded -polynomials. On the far right-hand side, the equation does not depend on the exponent , so reduction primitives can be precomputed in matrix form. When

,

the number of constraints equals the number of degrees of freedom equals . We then solve the system of linear equations by matrix inversion. In a matrix formulation, each becomes a -dimensional coefficient vector subject to Hermite reduction,

,

where indicates encoding to a -dimensional matrix. For each element of , Hermite reduction always terminates on . After reducing no more than of the one-forms, it becomes possible to find a coefficient vector such that , with certificate function . The integral over a complete period loses the exact differential, and we then obtain the Picard–Fuchs equation,

.

A Dihedral Case

The function DihedralPFDE[Hλϕ] operates on a more symmetrical range of plane curves. We start with a form such that can be written in reduced terms as

,

usually with . Action angle coordinates are canonical, so again by Hamilton's equations, we calculate that

.

After slightly more calculus, the angle variable may be eliminated, and we arrive at a set of three -polynomials,

,

,

,

the invariant polynomial data of . A vector collects one-forms,

,

and again we can write a pattern for all subsequent iterations of Hermite reduction,

.

If we introduce degrees

and

,

then we must also require that

.

This is the unique degree-bounding constraint where Hermite reduction becomes operable as

,

again with matrix encoding . For each element of , care must be taken in case the initial numerator satisfies

.

If so, we must precede Hermite reduction by expansion in partial fractions. Then recursive calculations always terminate on . After reducing no more than of the one-forms, it again becomes possible to find a coefficient vector such that

,

with certificate function . Again, the Picard–Fuchs equation,

,

follows immediately.

Certificates and Complexity

Both algorithms HEllipticPFDE[Hqp] and DihedralPFDE[Hλϕ] sow enough information to check coefficient vector against certificate function after evaluation. A successful computation is easy to validate by evaluating the expression , hopefully to a truth value 0, as is done in this Demonstration. It should be clear from the preceding discussion that computation of does not depend on computation of . However, components of are also computed by matrix multiplication in a relatively short time. Thus we are able to print and check the certificate functions without too much extra work.

How much work are we actually doing? And are we doing too much? If is the dimension of the vector space, then matrix inversion, matrix-encoded Hermite reduction and finding the nullspace all have an arithmetic complexity estimated roughly as . If partial decomposition becomes necessary, the dihedral algorithm can potentially slow to . Both polynomial figures are quite minimal, and compare well to complexity figures for other algorithms designed specifically for bivariate polynomials [4]. In practice, arithmetic complexity is somewhat misleading, and it is easy to notice exponential slowdown, especially when the size of the output coefficients and certificates begins to become unmanageably large.

More and More Interesting Geometry!

The new functions allow us to repeat and improve upon earlier calculations of "Deriving Hypergeometric Picard-Fuchs Equations", where elliptic Hamiltonians were measured by another special case of the hypergeometric differential equation

.

With the present Demonstration, we extend the range of geometries by three more interesting specimens featuring dihedral symmetry. Although we avoid Griffiths–Dwork reduction, the algorithms HEllipticPFDE[Hqp] and DihedralPFDE[Hλϕ] are not overly simple. They can produce many more interesting certificates!

The code commented out in the section Initialization Code contains numerous examples, but let us mention the Euler periods in particular. In a somewhat surprising success of input hacking, the function DihedralPFDE[Hλϕ] also enables analysis of periods of algebraic sphere curves. In fact, an earlier prototype of DihedralPFDE[Hλϕ] was used for the author's original derivation of differential equations associated to Goursat surfaces (cf. [5–7]). Quite amazingly, the real Euler period of the tetrahedral-symmetric, cubic sphere curves yet again takes a hypergeometric form,